Exact Path Integral for 3D Higher Spin Gravity

TL;DR

This paper computes the exact partition function of 3D higher spin gravity using localization, revealing connections to 2D CFT characters and confirming the Cardy formula at large central charge.

Contribution

It introduces a supersymmetric reformulation and exact localization computation for 3D higher spin gravity, providing explicit character expansions and positivity checks.

Findings

Partition function expressed in terms of 2D CFT characters

Coefficients of character expansion are positive integers

Large central charge limit reproduces Cardy formula

Abstract

Extending the works arXiv:1504.05991 and arXiv:1510.02142, we study three dimensional Euclidean higher spin gravity with negative cosmological constant. This theory can be formulated in terms of SL(N,C) Chern-Simons theory. By introducing auxiliary fields, we rewrite it in a supersymmetric way and compute its partition function exactly by using the localization method. We obtain a good expression for the partition function in terms of characters for the vacuum and primaries in 2D unitary CFT with W_N symmetry. We also check that the coefficients of the character expansion are positive integers and exhibit Cardy formula in the large central charge limit.